Convergence of Solutions for an Equation with State-Dependent Delay

Transcription

1 Joural of Mahemaical Aalysis ad Applicaios 254, doi:6jmaa2772, available olie a hp:wwwidealibrarycom o Covergece of Soluios for a Equaio wih Sae-Depede Delay Maria Barha Bolyai Isiue, Uiersiy of Szeged, Aradi erauk ere, H-672, Szeged, Hugary barham@mahu-szegedhu Submied by H L Smih Received February 4, 2 A resul of Smih ad Thieme shows ha if a semiflow is srogly order preservig, he a ypical orbi coverges o he se of equilibria For he equaio wih sae-depede delay x x fž xž rž xž, where ad f ad r are smooh real fucios wih fž ad f, we cosruc a semiflow which is moooe bu o srogly order preservig We prove a covergece resul uder a moooiciy codiio differe from he srog order preservig propery, ad apply i o he above equaio o obai geeric covergece 2 Academic Press INTRODUCTION I his paper we sudy he sae-depede delay equaio xž xž f xž r, r r xž, Ž where ad f ad r are smooh real fucios wih fž ad f Equaio Ž wih r appears i several applicaios; see 4,, 57, 9, 2, 22, 3 Over he pas several years i has become appare ha equaios wih sae-depede delay arise i several areas such as i classical elecrodyamics 59, i populaio models 2, i models of commodiy price flucuaios 3, 8, ad i models of blood cell producios 2 I he case r cosa cosider he space CŽr,, as a phase space Defie he closed parial order relaio i he followig way: wheever Ž s Ž s for all s r,, wheever ad X $35 Copyrigh 2 by Academic Press All righs of reproducio i ay form reserved 4

2 CONVERGENCE OF SOLUTIONS 4, ad wheever Ž s Ž s for all s r, Equaio Ž geeraes a semiflow H o he phase space The codiio f guaraees ha H is moooe, ha is, for every, i CŽr,, wih, H Ž, H Ž, holds for all I is also rue ha H is srogly order preservig Ž SOP, ha is, H is moooe, ad for every, i CŽr,, wih, here exis ad ope subses U, V of CŽr,, wih U ad V such ha H Ž, U H Ž, V The he resul of Smih ad Thieme 26, 29, 3 cocludes ha he omega limi se of a geeric poi of he phase space is coaied i he se of equilibria We remark ha aalogous resuls were obaied by Smih ad Thieme i 27, 28 for o-quasi-moooe fucioal differeial equaios, ha is, uder a weaker codiio ha f I he case r ržxž he siuaio becomes more difficul, because i is o obvious how o choose he phase space The quesios abou exisece, uiqueess, ad coiuous depedece of soluios of Eq Ž are also o sadard, see, eg, 25 I Secio 3 we show ha for suiable M ad A, he soluio of Eq Ž defies a semiflow H o he meric space coaiig Lipschiz coiuous fucios mappig M, io A, A wih meric dž, sup Ž s Ž s M s The cosa M is relaed o he maximum of r o A, A The resul of Smih ad Thieme is o applicable for his semiflow geeraed by Eq Ž, sice i does o have, i geeral, he SOP propery Ideed, cosider wo fucios ad i he phase space such ha Ž s Ž s A for all s i M, M ad Ž s Ž s for all s i M,, where Le U be a ope subse of he phase space wih U Clearly, here is a fucio U such ha s s for all s M, Le x ad x deoe he soluios of Eq Ž wih iiial fucio ad, respecively If we also have rž x Ž M, ad rž x Ž M, for all, he i is easy o see ha here exiss such ha H, Ž H, Ž Therefore, i his case H cao be SOP Our goal i his paper is o obai a covergece resul which is applicable for Eq Ž We observe ha H saisfies he followig propery H is moooe, ad for every ad i he phase space wih ad H, Ž H, Ž for all, here exis ad ope subses U, V of he phase space wih U ad V such ha H, Ž U H, Ž V This is why our aim is o prove a covergece resul for moooe semiflows havig he above propery isead of he SOP propery The followig assumpio seems o be crucial i achievig our goal If ad are i a compac ivaria subse of he phase space, he implies H, Ž H, Ž for all This codiio is saisfied for he semiflow geeraed by Eq Ž as well

3 42 MARIA BARTHA I he proofs of he moooiciy he hypohesis f ca be weakeed like i 27, 28, bu f seems o be crucial i he verificaio of he propery H, Ž H, Ž,, for all, i a compac ivaria subse wih The paper is orgaized as follows Secio 2 coais a geeral covergece resul, which is a modified versio of he covergece resul of Smih ad Thieme 26, 29 Secio 3 gives explici hypoheses o f ad r esurig he applicabiliy of our covergece resul for Eq Ž Secio 4 proves he geeral covergece resul 2 CONVERGENCE IN MONOTONE DYNAMICAL SYSTEMS I his secio we give he resul of Smih ad Thieme firs ad he he modified versio of his covergece resul I 26, 29 a meric space X is cosidered ad a closed parial order relaio o X For x ad y i X, x y is wrie wheever x y ad x y A semiflow is cosidered o X, ha is, a map :, X X, which saisfies: Ž i is coiuous, Ž ii Ž, x x for all x X, Ž ii Ž, Ž s, x Ž s, x for all, s, ad x X The orbi OŽ x of x X is defied by OŽ x Ž, x : 4 A poi x X is called a equilibrium poi if OŽ x x 4 The se of all equilibrium pois of is deoed by E The omega limi se Ž x of x X is defied by Ž x s, x Recall ha Ž x s is a oempy, compac, ivaria subse of X ad džž, x, Ž x aspro- vided OŽ x is a compac subse of X A poi x X is called a quasicoverge poi if Ž x E The se of all such pois is deoed by Q A poi x is called a coverge poi if Ž x cosiss of a sigle poi of E The se of all coverge pois is deoed by C I is supposed ha is moooe, ha is, for every x, y i X wih x y, Ž, x Ž, y holds for all I is assumed ha is srogly order preservig Ž SOP, ha is, is moooe, ad for every x, y i X wih x y, here exis ad ope subses U, V of X wih x U ad y V such ha Ž, U Ž, V Assume ha for each x i X, OŽ x has compac closure i X I is supposed ha for every x i X, Ž a here exiss a sequece Ž x i X saisfyig x x xž x x x for all iegers ad x x as, ad Ž b for he sequece Ž x wih he propery guaraeed by Ž a, Ž x has compac closure i X

4 CONVERGENCE OF SOLUTIONS 43 The he resul of Smih ad Thieme 26, 29 saes ha uder he above assumpios X I Q I C I paricular I Q is dese i X If we cosider Eq Ž, a difficuly arises: he semiflow geeraed by Eq Ž is o i geeral SOP Thus he resul of Smih ad Thieme is o applicable Our aim is o prove a covergece resul which is applicable for Eq Ž We observe ha he semiflow geeraed by Eq Ž has he followig propery is moooe, ad for x, y i X wih x y ad Ž, x Ž, y for all, here exis, ad ope subses U, V of X wih x U ad y V such ha Ž, U Ž, V We shall wrie x y if x y ad Ž, x Ž, y for all For wo subses a4 A ad B of X wih A B or A B or A B, we shall wrie a B or a B or a B We give some defiiios We say ha is mildly order preservig Ž MOP if i is moooe, ad for every x, y i X wih x y, here exis ad ope subses U, V of X wih x U ad y V such ha Ž, U Ž, V Noe ha he differece bewee he MOP ad SOP properies is ha we have oe more assumpio for he MOP propery, ha is, Ž, x Ž, y for all ad for all x, y i X wih x y Thus he MOP propery is weaker ha he SOP propery We say ha x X ca be approximaed from below Ž above i X if here exiss a sequece Ž x i X saisfyig x x xž x x x for all iegers ad x x as Sice he semiflow geeraed by Eq Ž is MOP, we are ieresed i provig a covergece resul for he moooe semiflow beig MOP isead of SOP Thus a aural quesio appears: Uder wha codiios ca we prove such a covergece resul? I Secio 4 we prove he followig heorem THEOREM 2 Cosider a meric space X wih a closed parial order relaio ad a semiflow o X Assume ha Ž A if x ad y are i a compac iaria subse of X, he x y implies x y, Ž A 2 is MOP, Ž A 3 each poi i X ca be approximaed eiher from below or from aboe ix, Ž A for each x i X, he orbi OŽ x 4 of x has compac closure i X, ad Ž A 5 for each x i X ad for each sequece x, which approximaes x eiher from below or from aboe ix, Ž x has compac closure i X The X I Q I C I paricular I Q is dese i X

5 44 MARIA BARTHA We meio ha he differeces bewee Theorem 2 ad he heorem of Smih ad Thieme are ha assumpio Ž A i Theorem 2 does o appear i he resul of Smih ad Thieme, he semiflow i Theorem 2 is MOP isead of SOP, ad he defiiio of approximaio i Theorem 2 differs from he oe used by Smih ad Thieme A examiaio of he proof of he resul of Smih ad Thieme shows ha due o assumpio Ž A he proof of Theorem 2 follows more or less he same lie as ha i 26, 29 3 CONVERGENCE OF SOLUTIONS FOR AN EQUATION WITH STATE-DEPENDENT DELAY I his secio we apply Theorem 2 for he differeial equaio wih sae-depede delay Ž Ž xž xž f x r xž, Ž 3 I order o achieve our goal, we eed o esablish hypoheses o f ad r ad choose a appropriae phase space esurig ha he codiios of Theorem 2 are saisfied Le, ad he fucios f ad r be i C Ž, wih fž ad fž x for all x Suppose ha here exiss a posiive cosa A such ha fž x x for all x A Le M max rž x x A, A ad assume ha rž x for all x A, A Se R max x fž y, ad lipž supž s Ž Ž x, ya, AA, A s : s, M,, s 4, where is i CŽM,, Le F be he map defied by he righ had side of Eq Ž 3, ha is, FŽ Ž fžžržž Observe ha F is o ecessary defied for all CŽM,, For CŽM,, wih Ž A, A, FŽ may o exis Therefore cosider he reracio : CŽM,, D defied by CŽM,, D, where D CŽM,, : A Ž A, for M 4 is a fucio mappig M, io A, A i he followig way: A for Ž A, Ž Ž Ž for A Ž A, Ž 32 A for Ž A Cosider he fucio F : CŽM,, defied by FŽ Ž Ž fž ŽrŽ Ž for all i CŽM,, ad he equaio xž F Ž x Ž 3

6 CONVERGENCE OF SOLUTIONS 45 Firs we show exisece, uiqueess, ad coiuous depedece of soluios of Eq Ž 3 ad he of soluios of Eq Ž 3 We say ha a fucio x : M, is a soluio of Eq Ž 3 o M,,, if here exiss such ha x : M, is coiuous o M,, i is coiuously differeiable o,, ad i saisfies Eq Ž 3 for all, We say ha a fucio x : M, A, A is a soluio of Eq Ž 3 o M, if x A, A for all M,, x : M, A, A is coiuous o M,, i is coiuously differeiable o,, ad i saisfies Eq Ž 3 for all We begi wih provig he followig claim CLAIM The fucio F : CŽM,, is coiuous ad saisfies he followig propery: here exis cosas a ad b such ha for all R ad for all, i CŽM,, wih lipž R, F Ž F Ž Ž a br Ž 33 Proof I is easy o check ha is coiuous; hus F is coiuous, ad for all, i CŽM,,, we have ad To show 33, we make he esimaios F Ž Ž F Ž Ž Ž Ž Ž 34 lipž Ž lipž Ž 35 Ž Ž Ž Ž Ž f Ž r Ž Ž Ž Ž f Ž r Ž Ž Ž Ž 36 The fucio f is locally Lipschizia because i is coiuously differeiable Thus here exiss L such ha Ž Ž f Ž r Ž Ž Ž f Ž r Ž Ž Ž L Ž r Ž Ž Ž Ž r Ž Ž Ž L Ž r Ž Ž Ž Ž r Ž Ž Ž L Ž r Ž Ž Ž Ž r Ž Ž Ž Ž 37

7 46 MARIA BARTHA Usig 35 ad he fac ha r is locally Lipschizia, beig coiuously differeiable, we obai ha here exiss N such ha Ž Ž Ž r Ž Ž Ž r Ž Ž R rž Ž Ž r Ž Ž Ž RN Ž Ž Ž Ž Ž 38 Combiig 37 wih 38 ad 34, we fid Ž Ž f Ž r Ž Ž Ž f Ž r Ž Ž Ž Ž LR N L Ž 39 We deduce from 36, 39, ad 34 ha F Ž F Ž Ž LR N L Ž L LNR Ž a br Observe ha he cosa a br depeds o ad i is idepede of The proof is complee For local ad global exisece we refer o Theorem of 25 Cosider CŽM,, Sice F : CŽM,, is coiuous, by Theorem of 25, for some, here exiss a soluio x : M, of Eq 3 hrough ; ha is, x is a soluio of Eq Ž 3 such ha x M, As here exis cosas C ad C2 such ha for all i CŽM,,, FŽ C C by Theorem of 25 2, he solu- io x ca be defied o M, For uiqueess we refer o Theorem 2 of 25 Cosider CŽM,, wih lipž, ad for some wo soluios y : M, ad z : M, of Eq Ž 3 hrough Sice F is coiuous o CŽM,, ad i saisfies propery Ž 33, by Theorem 2 of 25, we obai y Ž z Ž for all i M, For coiuous depedece of soluios of Eq Ž 3 we refer o Theorem 6 of 25 We defie he phase space X as he meric space of all real-valued coiuous fucios : M,A, A wih lipž, where he meric is obaied from max Ž s M s We iroduce he closed parial order relaio o X i he followig way: wheever Ž s Ž s for all s M,, wheever ad, ad wheever Ž s Ž s for all s M, To prove exisece, uiqueess, ad coiuous depedece of soluios of Eq Ž 3, we eed he followig proposiio

8 CONVERGENCE OF SOLUTIONS 47 PROPOSITION 3 For eery X he soluio x x : M, of Eq Ž 3 hrough saisfies xž A, A for all Proof Se sups : x Ž A, A for all, s 4 If belogs o he ierval of exisece, he eiher x A ad x or x A ad x Suppose x A ad x Sice r is posiive o A, A, we have x Ž rž xž A The moooiciy of f yields fž xž rž xž fž A The soluio x saisfies Eq Ž 3, ha is, x Ž Ž x Ž fž Ž x ŽrŽ Ž x Ž As x Ž A, A ad x ŽrŽ x Ž A, A, accordig o he defiiio of, we obai Ž x Ž x Ž ad Ž x ŽrŽŽ x Ž x ŽrŽx Ž Thus, x x fž xž rž xž The assumpio A fž A implies x Ž This is a coradicio A similar argume leads o a coradicio i he case x A ad x Therefore x A, A for all PROPOSITION 32 For eery X here is a uique soluio x x : M, A, A of Eq Ž 3 such ha x M, Proof Le X The here exiss a soluio x x : M, of Eq Ž 3 hrough By Proposiio 3, x A, A for all Hece Ž x x ad FŽŽ x FŽ x for all Thus, x : M, A, A is a soluio of Eq Ž 3 wih xm, Cosider a soluio y y : M, A, A of Eq Ž 3 such ha ym, Sice y Ž A, A for all, by he defiiio of, y Ž y, ad FŽ y FŽŽ y for all Thus y is also a soluio of Eq Ž 3 hrough ; herefore x Ž y Ž for all i he ierval of exisece PROPOSITION 33 Le,, X, x x : M, A, A be he uique soluio of equaio x Ž FŽ x such ha x M,, ad y y : M, A, A be he uique soluio of equaio y Ž FŽ y such ha ym, The here exiss a cosa C idepede of,, such ha C C xž yž e Ž e for all C Proof Sice x Ž A, A ad y Ž A, A for all, FŽ x FŽŽ x ad FŽ y FŽŽ y for all Thus x is he soluio of equaio x Ž FŽŽ x wih x M,, ad y is he soluio of equaio y Ž FŽŽ y wih y By Theorem 6 of 25 M,, here exiss C such ha we have he desired esimaio for x Ž y Ž We defie he map H by, X, x X, where x deoes he soluio of Eq 3 hrough, ad x is defied by x Ž s x Ž s for all s M,

9 48 MARIA BARTHA PROPOSITION 34 The map H is a semiflow o X, ha is: Ž i H is coiuous, Ž ii HŽ, for all X, Ž iii H, Ž HŽ s, H Ž s, for all, for all s, ad for all X Proof Proof of Ž i The coiuiy of H i he firs variable is obvious To prove he coiuiy of H i he secod variable, cosider he soluios x x ad y x of Eq Ž 3 hrough ad X, respecively Applyig Proposiio 33 wih, we obai ha here exiss a cosa Ž Ž C C such ha x y e for all Hece x y C e for all, which implies he coiuiy of H i he secod variable I is easy o see ha Ž ii ad Ž iii are saisfied PROPOSITION 35 The semiflow H is moooe, ha is, H, Ž H, Ž wheeer ad Proof Le, be i X wih, ad such ha A fž A For all i Ž,, cosider he equaio Ž Ž zž zž f z r zž Ž 3 Firs we show ha he soluio of Eq Ž 3 hrough exiss ad i is uique Cosider he reracio defied by Ž 32, he map FŽ Ž Ž fž ŽrŽ Ž for all CŽM,,, ad he equaio zž F Ž z Ž 3 Sice he fucio defied by he righ had side of Eq Ž 3 is coiuous, i saisfies propery Ž 33, ad here exis cosas C ad C2 such ha for all CŽM,,, FŽ C C 2, by Theorem of 25, we obai ha here exiss a uique soluio z : M, of Eq 3 wih z We prove ha z Ž A, A for all Se sups : z Ž A, A for all, s 4 If belogs o he ierval of exisece, he eiher z Ž A ad z Ž orz Ž A ad z Ž Suppose z Ž A ad z Ž Sice r is posiive, we have z Ž ržz Ž A The moooiciy of f yields fž z Ž r Ž rž z Ž fž A As z Ž A, A ad z ŽrŽz Ž A, A, we ifer Ž z Ž z Ž f z r z Ž ž ž // Ž Ž z Ž f z r z Ž AfŽ A

10 CONVERGENCE OF SOLUTIONS 49 The assumpio A fž A implies z Ž This is a coradicio A similar argume leads o a coradicio i he case z Ž A ad z Ž Therefore, z Ž A, A for all Hece Ž Ž Ž Ž z z ad F z F z for all Thus, z : M, A, A is a soluio of Eq Ž 3 hrough To prove uiqueess, cosider a soluio y : M, A, A of Eq 3 wih y Ž Sice y A, A for all, y is a soluio of Eq Ž 3 hrough Ž Ž Therefore z y for all M, Thus z : M, A, A is he uique soluio of Eq Ž 3 hrough Deoe by x ad y he soluios of Eq Ž 3 wih x ad y Proposiio 33 implies z y uiformly o compac subses of M, as Therefore, i order o coclude he moooiciy of H, i suffices o show ha x Ž z Ž for all ad for all i Ž, Fix a IfŽ Ž he z Ž xž So z Ž x Ž fž ŽrŽ xž fžžrž xž As ŽrŽxŽ ŽrŽxŽ, he assumpio f implies fžžrž xž fž ŽrŽ xž Cosequely, z Ž x Ž Hece here exiss so ha x Ž z Ž for all i Ž, The exisece of such a i he case Ž Ž follows immediaely from he coiuiy of z ad x Therefore, by way of coradicio we ca choose s such ha x Ž z Ž for s ad xž s z Ž s Clearly, xž s z Ž s O he oher had we have Ž Ž z Ž s z Ž s f z s r z Ž s Ž 3 As r is posiive o A, A, xžs rž xž s z Žs rž xž s z Žs rž z Ž s The assumpio f implies fž xžs rž xž s fž z Žs rž z Ž s Cosequely, Ž Ž Ž Ž xž s f x s r xž s z Ž s f z s r z Ž s Ž 32 Combiig Ž 3 wih Ž 32, we coclude z Ž s xž s This coradicio complees he proof Our ex goal is o prove ha if ad belog o some compac ivaria subse of X ad, he H This asserio follows immediaely from he followig lemma LEMMA 36 If x : A, A ad y : A, A are wo soluios of Eq Ž 3 ad x y, he xž yž for all

11 42 MARIA BARTHA Proof Se z x y We ge Ž Ž Ž Ž zž zž f x r xž f y r yž Ž Ž Ž Ž zž f x r xž f y r xž Usig he equaliy Ž Ž Ž Ž f y r xž f y r yž for all d fž u f Ž H fž su Ž s ds ds H we obai, for Ž f su s ds u for all u,, H Ž Ž Ž 33 zž zž f sx r xž Ž s y r xž ds Defie z rž xž H Ž Ž f sy r xž Ž s y r yž ds Ž Ž y r xž y r yž, Ž 34 Ž Ž y r xž y r yž H Ž Ž y s r xž Ž s r yž ds rž xž rž yž, Ž 35 H r xž r yž r sxž Ž s yž ds zž Ž 36 H Ž Ž až f sy r xž Ž s y r yž ds H Ž Ž y s r xž Ž s r yž ds H r sxž Ž s yž ds for all Ž 37

12 CONVERGENCE OF SOLUTIONS 42 ad H Ž Ž bž f sx r xž Ž s y r xž ds for all Ž 38 Combiig 34 wih 35 ad 36, i follows ha z saisfies he liear equaio Ž zž až zž bž z r xž for all, Ž 39 where a Ž ad b Ž are coiuous, bouded fucios defied by Ž 37 ad Ž 38 Moreover, b for all sice f Defie Ž Ž H až s ds H až s ds ze for all Muliplyig 39 by e, we ifer Ž b z r x e H až s ds for all Ž 32 Ž The defiiio of yields Ž Ž z r x r x e H rž xž až s ds for all Ž Thus saisfies he liear equaio Ž Ž cž r xž for all, Ž 32 where c Ž is a coiuous, bouded fucio defied by c Ž Ž H rž xž až s ds be for all, ad c Ž for all I order o show ha z Ž for all, i suffices o prove ha Ž for all Noe ha Ž for all M due o he uiqueess of soluios ad x y Se if : Ž s for all s 4 We claim ha Oherwise M We have Ž for all Therefore Ž for all The i follows by Ž 32 ha Ž rž xž for all, which usig he defiiio of, implies if ržxž : 4 O he oher had usig he assumpio ru for all u A, A, we obai if ržxž : 4 ržxž This coradicio shows ha Cosequely, Ž for all ad he lemma is proved Ž I he above proof i is impora ha he delay r depeds oly o x ad o o x The hypohesis f seems o be also crucial COROLLARY 37 Le B be a compac iaria subse of X, where iariace meas ha for ay B, here exiss a soluio x of Eq Ž 3 o wih x ad x B for all If, B wih, he x x for all The semiflow H has he followig propery

13 422 MARIA BARTHA PROPOSITION 38 If, X wih H, he here exiss such ha HŽ, H, Ž for all Proof Cosider, X wih H, ad he soluios x x : M, A, A ad y x : M, A, A of Eq Ž 3 hrough ad, respecively The moooiciy of H implies x Ž y Ž for all M Our aim is o prove ha x Ž y Ž for all 2 M The i follows ha x Ž s yž s for all s M, ad for all 3M, ha is, H, Ž H, Ž for all 3M Se z x y As i he proof of Lemma 36 i ca be show ha z saisfies a liear equaio Equaio Ž 34 holds as well, bu Ž 35 holds wheever he soluio y is differeiable, ha is, for M We also defie he fucios a Ž ad b Ž as i Ž 37 ad Ž 38 bu oly for M Equaio Ž 39 holds for M We Ž Ž H M až s ds defie ze for all M Isead of Ž 32 here we obai Ž Ž b z r x e H M až s ds for all M Ž 322 saisfies he equaio Ž Ž cž r xž for all M, Ž 323 Ž Ž Ž H rž xž až s ds where c is defied by c be for all M The assumpio H implies x2 M y 2 M, ha is, z2 M By he defiiio of, i follows ha Sice Ž 2 M for all M, here exiss u M,2M such ha Ž u Havig Ž rž xž for all M ad c Ž, by Ž 323, we deduce Ž for all M Therefore Ž Ž u for all u Hece x Ž y Ž for all 2 M The proof is complee Proposiio 35 ad Proposiio 38 imply ha H is MOP PROPOSITION 39 The semiflow H is MOP, ha is, i is moooe, ad for eery, i X wih H, here exis ad eighbourhoods U of ad V of such ha HŽ, U H, Ž V Proof We have already show he moooiciy of H i Proposiio 35 Cosider, i X wih H Proposiio 38 implies ha here exiss such ha H, Ž H, Ž There are eighbourhoods U of H, Ž ad V of H, Ž such ha U V By he coiuiy of H, Ž, here exis eighbourhoods U of ad V of such ha H, Ž U U ad H, Ž V V Cosequely, H, Ž U H, Ž V Hece H, Ž U H, Ž V as required As he proof of Proposiio 39 shows, we have also proved ha for all, i X wih H, here exiss ad eighbourhoods U of ad V of such ha H, Ž U H, Ž V Sice ca be chose 3M, by Proposiio 38, we coclude he followig propery of H

14 CONVERGENCE OF SOLUTIONS 423 Remark 3 If, X wih, he here exis eighbour- H hoods U of ad V of such ha H 3M, U H 3M, V Our ex goal is o prove ha each poi i X ca be approximaed eiher from below or from above i X I order o do his, we eed oe more propery of he semiflow H PROPOSITION 3 H, Ž If, X wih ad, he HŽ, Proof Cosider, X wih, ad he soluios x x : M, A, A ad y x : M, A, A of Eq Ž 3 hrough ad, respecively Suppose by way of coradicio ha here exiss s such ha x Ž y Ž for all s ad xž s yž s Clearly, xž s yž s We have xžs rž xž s yžs rž xž s yžs rž yž s sice r is posiive o A, A The assumpio f implies fžxžs rž xž s fž yžs rž yž s Therefore xž s yž s fž xžs rž xž s fž yžs rž yž s This coradicio complees he proof PROPOSITION 32 Each poi i X ca be approximaed eiher from below or from aboe ix Proof Le be i X Defie s mi A, s ad s max A, s for all s M, ad We prove ha ca be approximaed eiher from below by a subsequece of or from above by a subsequece of i X Clearly, ad are i X for all 4, ad ad as Firs we show ha for every 4, eiher or As for every, 4 H H, Proposiio 3 implies ha for every 4 ad, H, Ž H, Ž Hece i follows ha for every 4, eiher or Therefore here is a subsequece Ž H H k such ha eiher for all k 4 or for all k 4 k H H k Wihou loss of geeraliy assume ha for all k By 4 H k Remark 3, for all k 4, here is a eighbourhood Uk of such ha HŽ 3M, U HŽ 3M, k Sice as k, we obai ha k k for all k 4, here exiss l 4 such ha U k Cose- k l quely, for all k 4, here exiss l 4 such ha HŽ 3M, HŽ 3M, This implies ha for all k 4, here k l k exiss l 4 such ha H Now i is clear how o choose a k l k subsequece Ž k such ha i approximaes from above j I remais o show he followig PROPOSITION 33 For each i X, he orbi OŽ of has compac closure i X Furhermore, for each i X ad for each sequece Ž,

15 424 MARIA BARTHA which approximaes eiher from below or from aboe ix, has compac closure i X Proof The firs asserio follows by he ArzelaAscoli ` heorem usig he fac ha lipž x R for all M For he secod asserio oice ha, for all i X, is a oempy, compac, ad ivaria se, ad, for all i X, is coaied i he se of all fucios i X wih lipž R Thus, for a sequece Ž, which approximaes eiher from below or from above i X, is also icluded i he se of all fucios i X wih lipž R, which by he ArzelaAscoli ` heorem is compac Hece Ž is compac i X as well The se of equilibrium pois of X is E X : Ž s Ž for all s M,, ad fžž Ž 4 Noe ha for all, 2 E wih 2, we have eiher 2 or 2 Accordig o Claim 2 Ž Secio 4 a omega limi se cao coai wo pois, 2 such ha or The i follows ha for all i X, he se 2 2 E has a mos a sigle poi Therefore he se of quasicoverge pois Q coicides wih he se of coverge pois C Cosequely, Theorem 2 saes i his special case: THEOREM 34 Uder he aboe hypoheses o f ad r, X I C, ha is, I C is dese i X Noe ha i geeral X C Kriszi e al 3 have show he exisece of periodic orbis i he case r for cerai, f, ad r A similar resul is proved by Malle-Pare ad Nussbaum 23, 24, Kuag ad Smih 4, ad Ario e al i he case r ržxž wih egaive feedback codiio Kriszi ad Ario 2 have show ha here exiss a smooh disk of oquasicoverge pois for case r ržxž wih egaive feedback codiio A similar resul is expeced for Eq Ž 3 i he posiive feedback case 4 PROOF OF THEOREM 2 Hereafer we suppose ha assumpios Ž A Ž A 5 of Theorem 2 are saisfied The proof of Theorem 2 cosiss of he followig seps CLAIM Ž i If Ž T, x x for some T, he Ž x is a T-periodic orbi Ž ii If Ž, x x for belogig o some oempy ope subse of Ž,, he Ž, x p Eas Ž iii If Ž T, x x for some T, he Ž, x p Eas Proof The proofs of asserio Ž i ad Ž ii ca be foud i Smih 26, Theorem 2 To prove Ž iii cosider x Ž T, x Sice is MOP, here

16 CONVERGENCE OF SOLUTIONS 425 exis eighbourhoods U of x, V of Ž T, x, ad such ha Ž, U Ž, V As here exiss such ha Ž, x V for all Ž T, T, i follows ha Ž, x Ž, Ž, x for all Ž T, T Case Ž ii implies Ž, x p E as CLAIM 2 x y A omega limi se cao coai wo pois x ad y such ha Proof Suppose by way of coradicio ha here are x, y i Ž z such ha x y The x y because Ž z is a compac, ivaria subse of X As is MOP, here exis eighbourhoods U of x, V of y, ad such ha Ž, U Ž, V Choose such ha Ž, z U ad such ha Ž, z V Sice Ž, z V for all Ž 2 2 2, ad for some Ž,, i follows ha Ž, z 2 2 Ž, Ž, z Ž, Ž, z for all Ž, 2 2 By Claim Ž ii, Ž, z p E as Thus Ž z p ad x y, which is a coradicio A immediae cosequece of Claim 2 is ha a omega limi se cao coai a maximal miimal eleme Ž CLAIM 3 If a x ad x a a x, he x a Proof Cosider a Ž x ad Ž x a Suppose ha here exiss b Ž x such ha b a The b a Sice b ad a are i Ž x,we have obaied a coradicio o Claim 2 CLAIM 4 If x y,, Ž, x p, ad Ž, y k k k pask, he p E Proof Le x y Sice is MOP, here exis eighbourhoods U of x, V of y ad such ha Ž, U Ž, V Le be so small ha Ž s, x :s4 U ad Ž s, y :s4 V The Ž s, x Ž r, y for all s ad r i, Ž Thus, Ž s, x Ž, y for all s i, The moooiciy of implies Ž, Ž s, x Ž, y for all s i, k k ad for all large k Deoig r s, we ge Žr, Ž, x Ž, y k k for all r i, ad for all large k Passig o he limi as k, we ifer Ž r, p p for all r i, Replacig Ž s, x wih Ž, x i Ž ad arguig as above, we obai p Ž r, p for all r i, Thus, Ž r, p p for all r ad herefore, for all r, so p E CLAIM 5 If x y he x y E Proof Cosider p Ž x Ž y The here exiss a sequece Ž k such ha ad Ž, x p as k ŽŽ, y is a sequece i k k k

17 426 MARIA BARTHA he compac se OŽ y By passig o a subsequece if ecessary, we ca assume ha Ž, y k q as k The moooiciy of implies Ž, x Ž, y k k for all iegers k Leig k, we fid ha p q The case p q coradics Claim 2, sice p, q Ž y Hece p q ad by Claim 4, p E CLAIM 6 Le K ad K 2 be compac subses of X saisfyig K K 2 The here are ope ses U ad V, wih K U ad K 2 V, ad, such ha Ž s, U Ž, V for all ad for all s Proof Fix a x i K Sice is MOP, for each y K 2, here exis eighbourhoods U of x, V of y, ad such ha Ž, U Ž, V y y y y y for all As K is compac ad V 4 y 2 y y K is a ope cover of K 2,we 2 may choose a fiie subcover, K 2 i V y, where yi K 2 for all i i Se V i V y, U i Uy ad max i y The i i i Ž, U Ž, U y, Vy for all ad for all i I follows i i ha Ž, U Ž, V for all Deoe V V ad U U x x o empha- size he depedece of hese ope ses o he poi x K Similarly, x We have obaied ha for each x K, here exis eighbourhoods U Ž Ž x of x, Vx of K 2, ad x such ha, Ux, Vx for all x Agai, as U 4 x x K is a ope cover of K, we may exrac a fiie subcover, K m U i x, where xi K for all i m Se U i m m U, V V, ad max Sice Ž, V Ž, V i xi i x i i m xi xi ad Ž, U Ž, V x x for all ad for all i m, we coclude i i ha Ž, U Ž, V for all ad for all i m Thus, Ž, U x i Ž, V for all I order o obai he sroger coclusio of he claim, we observe ha, by he coiuiy of, for each x K, here exis ad a eighbourhood W of x such ha Ž, W x x x x U As W 4 x x K is a ope cover of K, we may choose a fiie subcover, K i m W x Deoe U i m Wx ad mi i m x If x U i i i ad s, he x W for some i Thus Ž s, x x i U Therefore, Ž, U U ad he Ž s, U U for all s I follows ha Ž s, U Ž, U Ž, V for all ad for all s CLAIM 7 If x y,, Ž, x a, Ž, y k k k bask ad a b, he OŽ a b Proof For u OŽ x, OŽ y wih u, defie JŽ u, supr :Ž, u,r 4 Our aim is o prove ha JŽ a, b Firs we verify wo properies of Ju, Ž Ž P J, u,, is moooe odecreasig i

18 CONVERGENCE OF SOLUTIONS 427 To show Ž P, i suffices o esablish JŽŽ, u, Ž, Ju, Ž for all We have Ž s, u for all s Ju, Ž The moooiciy of implies Žs, Ž, u Ž, for all s Ju, Ž ad Thus, JŽŽ, u, Ž, Ju, Ž for all Ž P 2 If uk k, uk OŽ x, k OŽ y, ad uk u, k, he lim sup Ju, Ž Ju, Ž k k k If Ju, Ž, he he asserio is obvious Assume Ju, Ž Suppose by way of coradicio here exiss such ha lim sup Ju, Ž Ju, Ž Le Ž k k k k i be a sequece i wih k as i such ha lim sup Ju, Ž lim Ju Ž, i k k k i k k Co- i i sequely, Ju, Ž Ju Ž, k k for all large i From he defiiio of i i Ju Ž, i follows ha Ž s, u for s Ju, Ž k k k k ad for all i i i i large i Leig i, we obai Ž s, u for s Ju, Ž, which coradics he defiiio of Ju, Ž Deoe lim JŽŽ, x, Ž, y, which exiss i, accordig o Ž P ByŽ P, we obai JŽ a, b Suppose JŽ a, b 2 For r JŽ a, b, Ž r, a b Moreover, Ž r, a b for r JŽ a, b Oherwise b Ž x, by he ivariace of Ž x, a Ž x, ad a b, i coradicio o Claim 2 We asser ha Ž r, a b for r JŽ a, b Ideed, by he ivariace of Ž x, Oa Ž x SoŽ r, a is i Ž x Ž y for r JŽ a, b ad b is also i Ž x Ž y As Ž x Ž y is compac ad ivaria, we deduce Ž r, a b for r JŽ a, b Se K Ž r, a :rjž a, b 4 K is compac ad K b Thus, Claim 6 implies ha here exis, ad ope ses U, V wih K U ad b V such ha Ž s, U Ž, V for all ad s Sice Ž, y b as k, here exiss a ieger k such ha Ž, y k k V for all k k AsŽ, x a as k, Žr, Ž, x Ž r, a k k uiformly i r, JŽ a, b as k Cosequely, here exiss k such ha Žr, Ž, x U for all k k ad for all r, JŽ a, b k We ifer Ž s, Žr, Ž, x Ž, Ž, y k k for all, for all s, for all k k maxk, k 4, ad for all r, JŽ a, b 2 O rearragig he argumes, we coclude Žr s, Ž, x Ž, y k k for all, for all k k ad for all s r JŽ a, b I follows ha JŽŽ 2, x, Ž, y JŽ a, b k k for all ad for all k k 2 Leig k, we obai JŽ a, b Bu JŽ a, b, which provides a coradicio Hece JŽ a, b The Oa b, ha is, Ž r, a b for all r Oherwise, as we have show above, we ge a coradicio o Claim 2 Moreover, Oa b Ideed, by he ivariace of Ž x, Ž r, a Ž x for all r, hus, Ž r, a Ž x Ž y for all r, ad b is also i Ž x Ž y The compacess ad ivariace of Ž x Ž y implies he desired asserio

19 428 MARIA BARTHA CLAIM 8 If u, X ad here exiss x Ž u such ha x Ž, he Ž u Ž Similarly, if here exiss x Ž u such ha Ž x, he Ž Ž u Proof Firs oe ha x Ž u, x Ž implies x Ž Ideed, we have x Ž u Ž ad y Ž u Ž for all y i Ž By he compacess ad ivariace of Ž u Ž, x y for all y Ž, ha is, x Ž Applyig Claim 6, we obai ha here exis ad eighbourhoods U of x ad V of Ž such ha Ž, U Ž, V Sice Ž V ad Ž is ivaria, Ž, U Ž As x Ž u, here exiss such ha Ž, u U Thus, Ž, u Ž The moooiciy of ad ivariace of Ž imply Ž s, u Ž for all s Hece Ž u Ž We asser ha Ž u Ž Suppose ha here exiss z i Ž u Ž Due o he fac ha z Ž ad z Ž, by Claim 3, we fid ha z Ž Similarly, sice Ž u z ad z Ž u, we ge z Ž u O he oher had x Ž implies x z, ad x Ž u implies x z, which is impossible Fially, Ž u Ž because of he compacess ad ivariace of Ž u Ž CLAIM 9 If x y,, Ž, x a, Ž, y k k k bask ad a b, he Ž x Ž y Proof Accordig o Claim 7, Oa b Hece Ž a b We asser ha Ž a b Suppose b Ž a The Ž a b, by Claim 3 Sice a Ž a ad a Ž x, Claim 8 implies Ž x Ž a This is impossible as Ž a Ž x Cosequely, Ž a b Due o he fac ha b is i Ž y, by Claim 8, we obai Ž a Ž y Sice every z Ž a belogs o Ž x as well, Claim 8 gives Ž x Ž y CLAIM If x y he eiher Ž a Ž x Ž y or Ž b Ž x Ž y E Proof If Ž x Ž y, accordig o Claim 5, we obai Ž x Ž y E If Ž x Ž y, he we may suppose ha here exiss q Ž y Ž x The oher case is reaed similarly There exiss a sequece Ž k such ha ad Ž, y q as k Sice ŽŽ, x k k k is a sequece i he compac se OŽ x, we may assume, by passig o a subsequece if ecessary, ha Ž, x k p as k The moooiciy of implies Ž, x Ž, y k k for all k Leig k, we ge p q We asser ha p q Ideed, if p q, he q Ž x, which is a coradicio Thus, by Claim 9, i follows ha Ž x Ž y CLAIM If x X ca be approximaed from below i X by a sequece Ž x, he here exiss a subsequece Ž x of Ž x such ha x x

20 CONVERGENCE OF SOLUTIONS 429 x for all iegers, wih x x as, saisfyig oe of he followig properies Ž a There exiss u E such ha Ž x Ž x u Ž x for all iegers ad lim dis u, Ž x b There exiss u E such ha Ž x u Ž x for all iegers If u E ad u Ž x, he u u Ž c Ž x Ž x E for all iegers A aalogous resul holds if x ca be approximaed from above i X Claim describes hree aleraives for he poi x : x ca be a coverge poi by Ž a, or x ca be a quasicoverge poi by Ž c, or x ca belog o he closure of he se of coverge pois accordig o Ž b Proof By Claim, here exiss a subsequece Ž x of x such ha eiher Ž x Ž x E for all iegers orž x Ž x for all iegers Cosider he case Ž x Ž x for all iegers, which is equivale o Ž x Ž x for all iegers, by he ivariace ad compacess of Ž x Ž x I follows ha Ž x Ž x for all iegers Ideed, if here exiss such ha Ž x Ž x, he Ž x Ž x for all, which is a coradicio Se y : y lim y, y Ž x 4 Ž x is oempy due o he fac ha Ž y is a moooe sequece i he compac se Ž x We claim ha cosiss of a sigle eleme, ha is, u 4 Ideed, if here are y ad u i so ha y y ad u u as, where y, u Ž x, he Ž x Ž x implies y u ad u y for all iegers Leig, we ifer y u ad u y, ha is, y u We claim ha u E Cosider y Ž x The y u as By he coiuiy of, Ž, y Ž, u as Sice Ž, y Ž x by he ivariace of Ž x, we obai Ž, y u as Thus, Ž, u u for all I follows from he defiiio of ad he compacess of x ha lim disžu, Ž x Ž Fially, Ž x Ž x for all iegers implies u Ž x If u Ž x, he by Claim 3, we ge Ž x u, which is case Ž a

21 43 MARIA BARTHA Suppose u Ž x The u Ž x, ha is, u Ž x by he ivariace ad compacess of u 4 Ž x Claim 6 implies ha here is a eighbourhood W of Ž x ad such ha u Ž, u Ž, W for all There exiss such ha Ž, x W By he coiuiy of Ž,, here is a ieger such ha Ž, x W for all Cosequely, u Ž, Ž, x for all ad for all Leig, we obai u Ž x for all Sice Ž x Ž x for all large k, i follows ha Ž x k u Thus, u Ž x Ž x u, which is a coradicio Cosider he case Ž x Ž x E for all iegers As x x, Claim implies ha eiher Ž x Ž x E for all iegers, which is case Ž c, or Ž x Ž x for all iegers Sup- pose Ž x Ž x for all iegers Le u Ž x Ž x E Cosequely, u Ž x Arguig exacly as above, we obai ha here exis a ieger ad such ha u Ž, x 2 for all ad for all The u Ž x 2 for all Sice u Ž x, by Claim 3, Ž x u for all Fially, if u E ad u Ž x, he arguig as above, we fid ha u Ž x for all As Ž x u, i follows ha u u The ex resul gives some iformaio which sreghes he asserio cocerig case b of Claim CLAIM 2 Ž i I case Ž b we hae i addiio he followig properies: There exis a eighbourhood O of u,,, ad such ha Ž, O Ž, x for all Ž ii There is a eighbourhood U of x wih he followig propery: for each x U wih x x, here exis a eighbourhood V of x i U, a ieger N, ad T such ha iii x I C u Ž, V Ž, xn for all T Proof Proof of Ž i I case Ž b, we have u Ž x Thus Claim 6 implies ha here exis a eighbourhood W of Ž x, O of u, ad such ha Ž, O Ž, W for all There exiss such ha Ž, x W By he coiuiy of Ž,, i follows ha here exiss a ieger such ha Ž, x W The Ž, O Ž, x for all Proof of Ž ii We choose a eighbourhood U of x such ha Ž, U W Le x U wih x x Sice is MOP, here exis a eighbourhood V of x wih V U, N of x, ad such ha Ž, V Ž, N 2

22 CONVERGENCE OF SOLUTIONS 43 for all As here is a ieger N such ha x N, we ge Ž, V 2 N Ž, x for all ByŽ i,u Ž, u Ž, O Ž, W N 2 for all Due o he fac ha Ž, V Ž, U W, we obai u Ž, O Ž, V for all Hece u Ž, V Ž, x N for all T, where T 2 Proof of Ž iii Sice Ž x u, Ž, x N N u as Thus, by Ž ii, we obai Ž u for all i V Therefore, for all x U wih x x, we ge Ž x u ad x I C If we cosider he sequece Ž x, which approximaes x from below, he x I C for all large Hece x I C Now we are i he posiio o prove Theorem 2 Proof of Theorem 2 Suppose x X I Q The here exiss a sequece Ž y such ha y X Q ad y x as By assump- io Ž A 3 of Theorem 2, for each, y ca be approximaed eiher from below or from above i X Cosider he former case as he laer case is similar Usig Claim, we obai, by passig o a subsequece if ecessary, ha for each, here exiss a sequece Ž x m m such ha xm xm y for all iegers m ad xm y as m For each, y Q; herefore case Ž b of Claim mus hold Claim 2Ž iii implies ha for each, y I C Hece x I C, which complees he proof REFERENCES O Ario, K P Hadeler, ad M L Hbid, Exisece of periodic soluios for delay differeial equaios wih sae depede delay, J Differeial Equaios 44 Ž 998, J Belair, Populaio models wih sae-depede delays, i Lecure Noes i Pure ad Appl Mah, Vol 3, pp 6576, Dekker, New York, 99 3 J Belair ad M C Mackey, Cosumer memory ad price flucuaios o commodiy markes: A iegrodiffereial model, J Dyam Differeial Equaios Ž 989, M W Dersie, H M Gibbs, F A Hopf, ad D L Kapla, Bifurcaio gap i a hybrid opically bisable sysem, Phys Re A 26 Ž 982, R D Driver, Exisece heory for a delay-differeial sysem, Corib Differeial Equaios Ž 963, R D Driver, A wo-body problem of classical elecrodyamics: The oe-dimesioal case, A Phys 2 Ž 963, R D Driver, A fucioal differeial sysem of eural ype arisig i a wo-body problem of classical elecrodyamics, i Iera Symp Noliear Differeial Equaios ad Noliear Mechaics, pp , Academic Press, New York, R D Driver, The backwards problem for a delay-differeial sysem arisig i classical elecrodyamics, i Proc Fifh Iera Coferece o Noliear Oscillaios, Kiev, 969 ; Izdaie Is Ma Akad Nauk Ukrai SSR Kie 2 Ž 97, 3743 I Russia 9 R D Driver ad M J Norris, Noe o uiqueess for a oe-dimesioal wo-body problem of classical elecrodyamics, A Phys 42 Ž 967, 34735

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